MIT OpenCourseWarehttp://ocw.mit.edu 18.034 Honors Differential Equations Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.18.034 Recitation: April 7th, 2009 1. Volterra integral/Tautochrone example. 2. Suppose that f and g are piecewise continuous functions. Verify the fol-lowing properties of their convolution. (a) f ∗ g = g ∗ f. (b) If f ∈ C1, then f ∗ g is C1 and (f ∗ g)� = f� ∗ g. 3. Use the Heaviside expansion to find an expression for the rest solution to the equation y�� + 5y� + 6y = f(t). Verify your answer against the known solution in the case f(t) = 1. 4. (Heaviside superposition formula) Let T be a linear differential operator with time-independent coefficients. Suppose that f� is piecewise continu-ous, and f continuous, and let φ be the rest solution to T φ = h(t) (here h(t) denotes the unit step function). Express the rest solution to T y = f (t) in terms of φ. 5. Consider the differential equation y�� + y = h(t) − h(t − c) for c > 0. (a) Use the Laplace transform to find the rest solution. (b) Show that y and y� are continuous at t = c but y�� is not.
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